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In the archetypal Lindsay Davenport story, writes Simon Kuper, Qantas Airways refuses to let her take her rackets on to a plane. When Davenport asked why her main rival was allowed to board the same flight with rackets, the reply came: “Well, Martina Hingis needs her rackets.” To the airline, the 6ft 2in world’s number one female tennis player was just some pushy tourist. The Californian was reduced to smuggling her rackets on board, recounts Paul Fein in his book Tennis Confidential.

Six years later Davenport is back at number one. Next week she can seal the spot for 2005 at the season’s finale, the tour championships for the world’s eight best players in Los Angeles. Yet Qantas could easily make the same mistake again. At 29, after half a lifetime on tour, Davenport remains unappreciated. Partly this is because the world is shallow and demands that female tennis players be beautiful, but it’s not just that. Davenport has yet to prove she is as good as her ranking.

She was born for sport. Raised in a volleyball-mad family, she shared her wealthy southern Californian hometown of Palos Verdes with Pete Sampras and Tracy Austin. Nobody thought Davenport would emulate them, though. As a gawky teenager, she never expected to win even one professional tournament. She was plainly too slow. She joined the circuit at 15, made the top ten at 18, but always lost in Grand Slams.

The problem, she admitted, was chiefly psychological. Most of the season she played to public indifference. That was fine with her. Davenport would prefer it if tennis players got as much publicity as accountants. However, when fans woke up for the Grand Slams, they invariably supported her opponent. People had nothing against the gentle giantess, writer of thank-you notes to tournament directors. She just seemed a nobody, whereas Anna Kournikova was a goddess, Steffi Graf was a legend and the Williams sisters were feisty.

Davenport was never the player the crowd came to see. Feeling the energy against her in big matches, she usually lost. She explains: “It was a form of self-destruction. If things got very exciting, I’d look at my opponent and think, ‘You really want to win? Take it.’” Her coach, Robert van’t Hof, said the reason she won gold at the Atlanta Olympics was that the crowds there would root for any American, even Benedict Arnold. Only later did Billie Jean King, then the American Fed Cup captain, convince Davenport that trying to win was good for the psyche.

Davenport’s game improved. She applied “portion control” to pancakes and chocolate cake, and did aerobics and basketball exercises to speed her footwork. To borrow a phrase from the soccer coach Berti Vogts, she still “dances like a refrigerator” on court. However, her serve is so hard she regularly hits aces even on clay and the only woman’s groundstroke bigger than Davenport’s forehand may be Davenport’s backhand. She brings the power play of men’s tennis to the women’s tour. For 18 months around the millennium, she was unquestionably the world’s best, winning three Grand Slams.

Since January 2000, though, she has won none. Now the old lady of the tour, she may never win another. This, to many fans, marks her as practically a loser. Most fans only watch tennis during Grand Slams when they see Davenport lose. There is a belief that any other top player at her best will beat her.

Of course this is unfair. However, Davenport’s failure to win Grand Slams is curious. Last month she won her 50th title and Mary Pierce says: “Lindsay’s definitely the greatest player right now playing.” Yet in the biggest matches, her attitude retains a trace of the old, “Take it.” Tennis, like chess or boxing, is largely a battle of two people’s wills. Faced with hypercompetitive women – Justine Henin or Venus Williams – Davenport doesn’t seem desperate enough. After losing a thrilling Wimbledon final this summer, she was characteristically kind: “It was just amazing to me how Venus, every time the chips were down, played unbelievable. Every time I got up, she took it away from me.” Unfortunately nobody ever says the same thing about Davenport. “The killer instinct is not very strong in her,” says the former Wimbledon champion Virginia Wade.

The problem explains itself when you hear Davenport speak. Instantly it becomes impossible to imagine her as a top-class athlete. Smiling, polite and eager to please, she spends her sparsely attended press conferences putting herself down. “The other night I saw myself on TV and ran out of the room” is a typical example. It doesn’t appear an act. Enchantingly, Davenport seems unaware that she is a genius.

When not deprecating herself, she deprecates her job. To the little band of American journalists who follow her around the world, she confides that she finds long stays in Europe difficult: “I’m much older now, I’m married, my husband can’t travel, I have other priorities.” Other priorities? You’re number one in the world, Lindsay! Sometimes it seems she only shows up at tournaments because saying no would be rude. Her husband, an investment banker and former all-American tennis player, had to talk her into playing this year’s French Open. Recently she almost retired from tennis, before realising she would miss it too much.

Great athletes are dissatisfied people, always talking about future goals. There is one exception. “I’ve been fortunate for 12 years to have an extraordinary amount of success,” says Davenport. “I was different from a lot of girls growing up. I never possessed that self-confidence they have. And look at me – I don’t know – got to number one.” That seems enough for her, if not for the world.